Cosine of Complement equals Sine

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Theorem

$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$

where $\cos$ and $\sin$ are cosine and sine respectively.

That is, the sine of an angle is the cosine of its complement.


Proof 1

\(\ds \map \cos {\frac \pi 2 - \theta}\) \(=\) \(\ds \cos \frac \pi 2 \cos \theta + \sin \frac \pi 2 \sin \theta\) Cosine of Difference
\(\ds \) \(=\) \(\ds 0 \times \cos \theta + 1 \times \sin \theta\) Cosine of Right Angle and Sine of Right Angle
\(\ds \) \(=\) \(\ds \sin \theta\)

$\blacksquare$


Proof 2

\(\ds \map \cos {\frac \pi 2 - \theta}\) \(=\) \(\ds \map \cos {\theta - \frac \pi 2}\) Cosine Function is Even
\(\ds \) \(=\) \(\ds \map \sin {\theta - \frac \pi 2 + \frac \pi 2}\) Sine of Angle plus Right Angle
\(\ds \) \(=\) \(\ds \sin \theta\)

$\blacksquare$


Proof 3

\(\ds \map \cos {\dfrac \pi 2 - \theta}\) \(=\) \(\ds \frac 1 2 \paren {e^{i \paren {\frac \pi 2 - \theta} } + e^{-i \paren {\frac \pi 2 - \theta} } }\) Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {e^{i \frac \pi 2} e^{-i \theta} + e^{-i \frac \pi 2} e^{i \theta} }\) Exponential of Sum: Complex Numbers
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\paren {\map \cos {\frac \pi 2} + i \, \map \sin {\frac \pi 2} } e^{-i \theta} + \paren {\map \cos {-\frac \pi 2} + i \, \map \sin {-\frac \pi 2} } e^{i \theta} }\) Euler's Formula
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {i e^{-i \theta} - i e^{i \theta} }\) Cosine of Right Angle, Sine of Right Angle, Cosine Function is Even, Sine Function is Odd
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {e^{i \theta} - e^{-i \theta} }\) Definition of Imaginary Unit
\(\ds \) \(=\) \(\ds \sin \theta\) Euler's Sine Identity

$\blacksquare$


Proof 4

ComplementaryAngles.png

Let $\angle xOP$ and $\angle QOy$ be complementary.

Then:

$\angle xOP = \angle QOy$

Hence:

the projection of $OP$ on the $x$-axis

equals:

the projection of $OQ$ on the $y$-axis.

Hence the result.

$\blacksquare$


Also see


Historical Note

The result Cosine of Complement equals Sine was discovered and documented by Varahamihira in the $6$th century CE.


Sources

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